Solve each inequality in Exercises 86–91 using a graphing utility. (x - 4)/(x - 1) ≤ 0

Verified step by step guidance

1. Identify the critical points by setting the numerator and denominator equal to zero: \(x - 4 = 0\) and \(x - 1 = 0\).
2. Solve these equations to find the critical points: \(x = 4\) and \(x = 1\).
3. Use these critical points to divide the number line into intervals: \((-\infty, 1)\), \((1, 4)\), and \((4, \infty)\).
4. Test a point from each interval in the inequality \(\frac{x - 4}{x - 1} \leq 0\) to determine where the inequality holds true.
5. Consider the behavior at the critical points: \(x = 1\) is a vertical asymptote, and \(x = 4\) is where the expression equals zero.

Key Concepts

Inequalities

Inequalities express a relationship where one side is not necessarily equal to the other, using symbols like ≤, ≥, <, or >. In this case, the inequality (x - 4)/(x - 1) ≤ 0 indicates that we are looking for values of x that make the expression less than or equal to zero. Understanding how to manipulate and solve inequalities is crucial for finding the solution set.

Rational Functions

A rational function is a ratio of two polynomials, such as (x - 4)/(x - 1). Analyzing rational functions involves identifying their domain, asymptotes, and intercepts. For this inequality, we need to determine where the function is zero or undefined, which will help us understand the intervals to test for the inequality.

Graphing Utilities

Graphing utilities are tools that allow users to visualize mathematical functions and inequalities. By graphing the rational function (x - 4)/(x - 1), we can easily identify where the function is below or equal to zero. This visual representation aids in determining the solution set for the inequality by observing the regions of the graph that satisfy the condition.